BY CLAUDE IRWIN PALMER, A.B. ASSOCIATE PROFESSOR OF MATHEMATICS ARMOUR INSTITUTE OF TECHNOLOGY AND DANIEL POMEROY TAYLOR, A.M. EDITED BY GEORGE WILLIAM MYERS, Ph.D. PROFESSOR OF THE TEACHING OF MATHEMATICS SCHOOL OF EDUCATION SCOTT, FORESMAN AND COMPANY NEW YORK PREFACE The purpose of this book is to present the subject of solid geometry so as to emphasize some of its applications as well as to give a thorough training in logical reasoning. To accomplish this the authors have continued the same method of presentation that characterizes their Plane Geometry. The aim of the authors in the preparation of this text has been to present the subject in a logical manner that is comprehensible to the youthful mind, and to vitalize the subjectmatter — making it both interesting and useful through a wide range of practical applications. Some of the features peculiar to the Solid Geometry are outlined in the following paragraphs: 1. In Chapter VI, the theorems on the relations of lines to planes and of planes to planes are carefully grouped so as to make them more easily.comprehended by the student. 2. Distinct advantage is gained by combining certain related subjects, thus securing brevity and simplicity of treatment. Prisms are combined with cylinders, pyramids with cones, and polyedral angles with spherical polygons. 3. In the theorems regarding the areas and volumes of solids, and requiring a use of limits, great care is taken to make the treatment logical and still keep it within the grasp of the average student. 4. The large number and great variety of exercises are carefully distributed, and range from some quite elementary to others of considerable difficulty. This enables the teacher to adapt the book to any group of students, whether in a classical or a technical high school. The abstract exercises give mental training and application of the basic theorems, while the practical exercises are used to correlate geometric facts with real life. Many of the exercises involve an application of arithmetic and algebra to geometry. In addition to the four points outlined above, attention is called to the following features: While many theorems are proved in full, the proof of others is given only in part. In still others the method of proof is suggested only or the work is left entirely to the student. Thus a middle course is adopted in the use of the suggestive method. The Prismatoid Formula, one of the most powerful theorems of solid geometry, is proved in a simple direct manner, and is followed by various applications. However, this may be omitted and in no way interfere with the proofs of later theo rems. The more important theorems are printed in heavy type. This is in accordance with the recommendation of the Committee of Fifteen, whose report, as well as various other committee reports, have been given careful consideration in the preparation of the entire work. For convenience, a combination ruler and protractor accompanies each book. Acknowledgment is due to the McGraw-Hill Book Co., Inc., for permission to use exercises from Palmer's Practical Mathematics, in which are gathered numerous exercises from the author's many years of experience in teaching practical students. C. I. PALMER. Chicago, September, 1918. CONTENTS PAGE Introduction CHAPTER VI. STRAIGHT LINES AND PLANES Relative Positions of Lines and Planes CHAPTER VII. POLYEDRONS, PRISMS, CYLINDERS Congruent and Equivalent Solids |